Introduction to coupled-cluster and equation-of - Q

Introduction to coupled-cluster and
equation-of-motion methods in Q-Chem
Evgeny Epifanovsky
August 9, 2014
Coupled-cluster theory for the ground state
Transition state theory
Definition
Transition state complex is the activated complex corresponding to a
maximum on an energy path along the coordinate of an elementary
reaction.
The transition state corresponds to a saddle point on the potential energy
surface. Hence, one negative eigenvalue of the Hessian, a matrix of partial
second derivatives.
Temperature dependence of the reaction rate is given by the Arrhenius
equation:
k(T) = Ae−Ea /RT
Study of a chemical reaction
C2 H4 + •CH3 → •CH2 CH2 CH3
Reactants
Transition state
Product
reac_c2h4_ch3.xyz
reac_c3h7_ts.xyz
reac_c3h7.xyz
$rem
method = ccsd
basis = cc-pvdz
gui = 2
$end
Study of a chemical reaction
C2 H4 + •CH3 → •CH2 CH2 CH3
Reactants
Transition state
Product
reac_c2h4_ch3.xyz
reac_c3h7_ts.xyz
reac_c3h7.xyz
B3LYP
6-31G*
-118.422822
-2.3
-118.419142
0.0
-118.470852 a.u.
-32.4 kcal/mol
CCSD
cc-pVDZ
-118.055116
-6.9
-118.044096
0.0
-118.100818 a.u.
-35.6 kcal/mol
CCSD(T)
cc-pVDZ
-118.067803
-6.2
-118.057938
0.0
-118.112890
-34.5 kcal/mol
1 a.u. = 627.51 kcal/mol
Equation-of-motion theory for excited states
Equation-of-motion method (EOM)
Based on the couple-cluster wave function for the ground state
|ΨEOM
i = Rm |ΨCC i = Rm eT |Φ0 i
m
T = T1 + T2 + · · ·
T1 =
X
tia a† i
T2 =
ia
1 X ab † †
tij a b ji
4
ijab
Rm = rm,0 + Rm,1 + Rm,2 + · · ·
Electronic correlation is folded into a similarity transformed Hamiltonian
¯ = e−T HeT
H
Wave functions satisfy the following equations
¯ − E CC |Φ0 i = 0
hΦµ |H
¯ − E EOM |Rm Φ0 i = 0
hΦµ |H
m
Excited state eigenfunctions are found by diagonalizing the similarity
transformed Hamiltonian in the Fock space.
Equation-of-motion method (EOM)
Variations of the EOM methods depend on the character of operator R.
Here is some of them:
EOM-EE (Excitation energy)
EOM-SF (Spin-flip)
EOM-IP (Ionization energy)
EOM-EA (Electron affinity)
Equation-of-motion method (EOM)
Benefits of EOM methods:
I Black box approach: no need to manually select active molecular
orbitals.
I Target states of different character (local, Rydberg, charge transfer)
are found in a single calculation.
I Reference state can be chosen based on convenience, not relevance to
the target states.
Q-Chem capabilities:
I Analytic gradients for geometry optimization.
I Excited state and transition properties.
I Minimization of potential surface crossings.
Important note:
I EOM solutions are not orthogonal, but biorthogonal because the
similarity transformed Hamiltonian is not Hermitian.
hΨCC LI |RJ ΨCC i = δIJ
Problems
1. Charge transfer state in Cs + triazine.
2. Character of excited state in diazirine (CH2 N2 ).
Cs + triazine
Q-Chem input file
triazine.in:
$rem
jobtype = sp
method = eom-ccsd
basis = 3-21g*
scf_algorithm = diis
scf_guess = core
max_scf_cycles = 200
n_frozen_core = fc
ee_singlets = [8,8]
$end
Diazirine
In a two-step process S1 state is excited first, then S2 . Both
transitions must be allowed, and S1 state must have some life
time.
The energy of photons in a REMPI(2+1) experiment is
3.9 eV. The problem is to find which states are accessible in
the experiment and describe their character.
How to solve this problem?
I
Compute the vertical excitation energies of the lowest states. Which
ones agree with the photon energy in the experiment?
I
Calculate the properties of the respective electronic transitions and
verify that the transitions are allowed (i.e. have non-zero oscillator
strength).
Diazirine
Q-Chem inpute file
diazirine.in:
$rem
jobtype = sp
method = eom-ccsd
basis = 6-31g*
ee_singlets = [2,2,2,2]
!cc_state_to_opt = [4,1]
cc_trans_prop = true
$end

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